Find the distance between the point ${(1, -3)}$ and the line $\enspace {y = -\dfrac{1}{3}x + 4}\thinspace$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Solution: First, find the equation of the perpendicular line that passes through ${(1, -3)}$ The slope of the blue line is ${-\dfrac{1}{3}}$ , and its negative reciprocal is ${3}$ Thus, the equation of our perpendicular line will be of the form $\enspace {y = 3x + b}\thinspace$ We can plug our point, ${(1, -3)}$ , into this equation to solve for ${b}$ , the y-intercept. $-3 = {3}(1) + {b}$ $-3 = 3 + {b}$ $-3 - 3 = {b} = -6$ The equation of the perpendicular line is $\enspace {y = 3x - 6}\thinspace$ We can see from the graph (or by setting the equations equal to one another) that the two lines intersect at the point ${(3, 3)}$ . Thus, the distance we're looking for is the distance between the two red points. The distance formula tells us that the distance between two points is equal to: $\sqrt{( x_{1} - x_{2} )^2 + ( y_{1} - y_{2} )^2}$ Plugging in our points ${(1, -3)}$ and ${(3, 3)}$ gives us: $\sqrt{( {1} - {3} )^2 + ( {-3} - {3} )^2}$ $= \sqrt{( -2 )^2 + ( -6 )^2} = \sqrt{40} = 2\sqrt{10}$ The distance between the point ${(1, -3)}$ and the line $\thinspace {y = -\dfrac{1}{3}x + 4}\enspace$ is $\thinspace2\sqrt{10}$.